In conventional seismic monitoring one or more seismic sources, such as airguns, vibrators or explosives are activated and generate sufficient acoustic energy to penetrate the earth. Reflected or refracted parts of this energy are then recorded by seismic receivers such as hydrophones and geophones.
In passive seismic monitoring there is no actively controlled and triggered source. The seismic energy is generated through so-called microseismic events caused by subterranean shifts and changes that at least partially give rise to acoustic waves which in turn can be recorded using the known receivers. These microseismic events may be the result of human activity, such as pumping a pressurized fluid into a subterranean location to create a hydraulic fracture. Passive seismic monitoring has some similarity to the study of earthquakes, in that the time and location of a seismic event is not known beforehand, while the obvious difference is that an earthquake is a much more energetic and spatially distributed seismic event.
It is well known that the characteristics of a seismic event can be expressed as the moment tensor and many textbooks and papers have accepted that the moment tensor is an appropriate way to describe a small seismic source. The moment tensor is a three by three symmetric matrix of values which give the magnitudes of all the possible force couples. Its name arises because it has the units of force times distance, hence a moment.
At a seismic event there is a localized, transient, failure of the constitutive law of elasticity. The difference between the model stress calculated from the constitutive law of elasticity and the true stress is referred to as the stress glut: thusσglut=σmodel−σtrue  (E1)
The moment tensor M is the integral of the rate of change of the stress glut over the volume and time period in which the seismic event occurs:
                    M        =                                            ∫                              0                -                            T                        ⁢                                          ∫                                  V                  S                                            ⁢                                                                    ∂                                          σ                      glut                                                                            ∂                    t                                                  ⁢                                                                  ⁢                                  ⅆ                  V                                ⁢                                                                  ⁢                                  ⅆ                  t                                                              =                                    ∫                              V                S                                      ⁢                                          [                                  σ                  glut                                ]                            ⁢                                                          ⁢                              ⅆ                V                                                                        (                  E          ⁢                                          ⁢          2                )            where T is the duration of the event (which we assume occurs at time zero),VS is the volume of the source (i.e. seismic event), and[σglut] is the change (saltus) of the stress glut during the event.
Provided the seismic event is small compared with the seismic wavelength and is of short duration (both of which are normally the case for a microseismic event) it can be regarded as a point source and the stress glut can be written asσglut(x,t)=MH(t)δ(x−xS)  (E3)where H(t) is the Heaviside step function, δ(x) is the three-dimensional Dirac delta function, and xS is the location of the point source.
It is well known that for a small and short seismic event which can be regarded as a point source, the particle displacement, i.e. the displacement at one or more receivers, which may be observed as velocity or acceleration, is related to the moment tensor by the Green function:u(t,xR)=Gσ(t,xR,xS):M  (E4)where u is the particle displacement, Gσ is the third-order stress Green tensor and: is the scalar product, or contraction over two indices, of the tensors. The Green function can be calculated for the elastic Earth model using a known technique, e.g. ray theory or finite-difference method. Determining the moment tensor is then a linear inverse problem and methods to analyze a linear inverse problem are well known.
A description of the moment tensor to describe a point source seismic event is given in Section 3.3 of Aki and Richards (2002). Observing the moment tensor from seismic data is referred to in U.S. Pat. Nos. 5,377,104 and 7,647,183 and US published applications 2005/0190649 and 2010/0157730.
The subsequent step of interpretation of the moment tensor presents difficulties. It is desirable to decompose the moment tensor into other quantities in order to obtain a physical interpretation of the seismic event. There is a standard decomposition of seismic moment tensors, which has been used in earthquake studies and in microseismic monitoring. This is to decompose the moment tensor using its principal values (usually termed eigenvalues) and its axes, i.e.
                              M          ⁡                      (                                                                                T                    ^                                                                                        N                    ^                                                                                        P                    ^                                                                        )                          =                              (                                                                                T                    ^                                                                                        N                    ^                                                                                        P                    ^                                                                        )                    ⁢                      (                                                                                Λ                    T                                                                    0                                                  0                                                                              0                                                                      Λ                    N                                                                    0                                                                              0                                                  0                                                                      Λ                    P                                                                        )                                              (                  E          ⁢                                          ⁢          5                )            
The eigenvalues are ordered, i.e.ΛT≧ΛN≧ΛP  (E6)and the corresponding axes are known as the tension, {circumflex over (T)}, neutral, {circumflex over (N)}, and pressure, {circumflex over (P)}, axes. The principal axes describe the orientation of the seismic event, and the principal values the nature or type of seismic event. The standard method of interpretation is to transform these eigenvalues into parts that represent certain basic types of seismic event, e.g. one or more of an explosion, dipole(s), double couple(s), compensated linear vector dipole(s) (Knopoff and Randall, 1970; Jost and Herrmann, 1989; Riedesel and Jordan, 1989). Unfortunately the decomposition is always non-unique.
Many physical models generate the same moment tensor. The usual approach when implementing this standard method of decomposition is to choose the two simplest types of seismic event which are explosion and slip on a fault. However such decomposition leads to a value for explosion, a value for slip on a fault and a remainder which is not easy to interpret. More difficulty in interpretation arises if the geological formation in which the seismic event occurs is anisotropic or if the so-called displacement discontinuity (the movement of one portion of the formation relative to another) is not confined to slip on a fault but includes movement of one portion towards or way from another, for instance when a hydraulic fracture opens or closes.